cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

Comments

From Gus Wiseman, Apr 13 2019: (Start)
Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:
(21) (22) (32) (33) (43) (44) (54) (55)
(31) (41) (42) (52) (53) (63) (64)
(211) (221) (51) (61) (62) (72) (73)
(2111) (222) (2221) (71) (81) (82)
(2211) (22111) (2222) (22221) (91)
(21111) (211111) (22211) (222111) (22222)
(221111) (2211111) (222211)
(2111111) (21111111) (2221111)
(22111111)
(211111111)
(End)

Examples

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
                        1                          =  1
                      1 1 1                        =  3
                    1 1 . 1 1                      =  4
                  1 1 1 . 1 1 1                    =  6
                1 1 . 1 . 1 . 1 1                  =  6
              1 1 1 . 1 . 1 . 1 1 1                =  8
            1 1 . 1 . 1 . 1 . 1 . 1 1              =  8
          1 1 1 . 1 . 1 . 1 . 1 . 1 1 1            = 10
        1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1          = 10
      1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1        = 12
    1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1      = 12
  1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1    = 14
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1  = 14
(End)

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Column k = 2 of A325227.
Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.

Programs

  • Mathematica
    rule = 94; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* Michael De Vlieger, Dec 14 2015 *)

Formula

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.
a(n) = n+2 for n>1 and even.
a(n) = n+3 for n>1 and odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).
(End)

A071033 a(n) = n-th state of cellular automaton generated by "Rule 94" when started with a single ON cell.

Original entry on oeis.org

1, 111, 11011, 1110111, 110101011, 11101010111, 1101010101011, 111010101010111, 11010101010101011, 1110101010101010111, 110101010101010101011, 11101010101010101010111, 1101010101010101010101011, 111010101010101010101010111, 11010101010101010101010101011
Offset: 0

Views

Author

Hans Havermann, May 26 2002

Keywords

Comments

a(n) has length 2n+1.

Links

Crossrefs

This sequence, A118101 and A118102 are equivalent descriptions of the Rule 94 automaton.

Programs

  • Mathematica
    rule=94; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]], {k, 1, rows}]   (* Binary Representation of Rows *) (* Robert Price, Feb 21 2016 *)

Formula

Conjecture: a(n) = floor((1099*100^n + 9090)/990) + 1 for odd n > 1; a(n) = floor((1090*100^n + 10)/990) + 1 for even n > 1. - Karl V. Keller, Jr., Oct 25 2021

Extensions

Corrected by Hans Havermann, Jan 07 2012
Edited by N. J. A. Sloane, Oct 20 2015 at the suggestion of Michael De Vlieger and Kevin Ryde
More terms from Robert Price, Dec 06 2015

A118101 Decimal representation of n-th iteration of the Rule 94 elementary cellular automaton starting with a single ON cell.

Original entry on oeis.org

1, 7, 27, 119, 427, 1879, 6827, 30039, 109227, 480599, 1747627, 7689559, 27962027, 123032919, 447392427, 1968526679, 7158278827, 31496426839, 114532461227, 503942829399, 1832519379627, 8063085270359, 29320310074027, 129009364325719, 469124961184427
Offset: 0

Views

Author

Eric W. Weisstein, Apr 12 2006

Keywords

Examples

			From _Michael De Vlieger_, Oct 08 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right:
                    1 =       1
                  111 =       7
               1 1011 =      27
             111 0111 =     119
          1 1010 1011 =     427
        111 0101 0111 =   1 879
     1 1010 1010 1011 =   6 827
   111 0101 0101 0111 =  30 039
1 1010 1010 1010 1011 = 109 227
(End)

Links

Crossrefs

This sequence, A071033 and A118102 are equivalent descriptions of the Rule 94 automaton.

Programs

  • Mathematica
    clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[94, {{1}, 0}, 24]] (* Michael De Vlieger, Oct 08 2015 *)
  • Python
    print([(11*4**n + 10)//6 - 2*0**abs(n-1) if n%2 else (5*4**n + 1)//3 - 0**n for n in range(50)]) # Karl V. Keller, Jr., Sep 10 2021

Formula

a(0)=1, a(1)=7, a(n odd) = (10+11*4^n)/6, a(n even) = (1+5*4^n)/3.
From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)
a(n) = (12-(-4)^n-8*(-1)^n+21*4^n)/12 for n>1.
a(n) = 17*a(n-2) - 16*a(n-4) for n>5.
G.f.: -(2*x+1)*(16*x^4-5*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)

A265284 Total number of ON (black) cells after n iterations of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 8, 14, 20, 28, 36, 46, 56, 68, 80, 94, 108, 124, 140, 158, 176, 196, 216, 238, 260, 284, 308, 334, 360, 388, 416, 446, 476, 508, 540, 574, 608, 644, 680, 718, 756, 796, 836, 878, 920, 964, 1008, 1054, 1100, 1148, 1196, 1246, 1296, 1348, 1400, 1454
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A118102.

Programs

  • Mathematica
    rule = 94; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]

Formula

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (2*n^2+12*n-(-1)^n+1)/4 for n>0.
a(n) = (n^2+6*n)/2 for n>1 and even.
a(n) = (n^2+6*n+1)/2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
G.f.: (1+2*x-x^4) / ((1-x)^3*(1+x)).
(End)
Showing 1-4 of 4 results.

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